Castigliano’s Second Theorem Gandhinagar Institute Of Technology Active Learning Assignments Structural Analysis-II (2150608) Castigliano’s Second Theorem Prepared By – Devanshu Tailor (120120106107) Roshan Aswani (120120106073) Krunal Patel (120120106131) Guided By – Prof. Sandip Kapdiya Prof. Jay Parmar

Castgliano’s second theorem : It states, “ If U is the total strain energy stored up in a framework in equilibrium under an external system of forces, its magnitude is always a minimum.” If U is the total strain energy stored in a body, the partial derivative of internal energy with respect to a load applied at a point where deflection is zero. ……… (1)

This theorem is also known as the Principle of least work or minimum strain energy. The structure will deform due to the applied loads by the minimum amount possible, so that the work done on it or the internal energy stored is a minimum.

For the members in bending : (Beams and rigid jointed frames) …….. (2)

For the members in axial loading : (pin jointed truss) …… (3)

Example 1: Determine reaction at prop for a propped cantilever beam using Castigliano’s second theorem as shown in figure.

Segment Origin Limits Mx (ðMx/ðVb) BC B 0 to 3 Vb.x – 10(x²/2) x CA C 0 to 2 Vb(x+3)-30(x+1.5)-20x-10(x²/2) (x+3)

9VB – 101.25 + 41.66 VB – 9VB – 620 – 173.33 – 60 = 0 .’. VB = 22.91 kN .’. VA = Total load – VB = (10 × 5 + 20 ) – 22.91 = 47.09 kN MA = 22.91 × 5 – 10 × 5 × 2.5 – 20 × 2 = - 50.45 kN.m MC = 22.91 × 3 - 10 × 3 × 1.5 = 23.73 kN.m

Example 2: Determine reaction at supports for a continues beam using Castigliano’s second theorem as shown in figure.

S.I = (2+1+1)-3=1 Consider as a redundant and find out other reactions by statics. Taking moments @ A .’. VB × 6 + Vc × 10 = (30 ×6 ×3) + (80 ×8) VB × 6 + Vc × 10 = 1180 .’. VA = Total load - VB – Vc = (180 + 80) - VB –(118-0.6 VB ) =142-0.4 VB

Segment Origin Limits Mx (ðMx/ðVb) AB A 0 to 6 142x-0.4VB x-30x(x/2) -0.4x CD DB C D 0 to 2 118x-0.6VBx (118x-0.6VB)(X+2)-80x =118x-0.6VBX+236 -1.2 VB-80X =38x-0.6 VBx+236-1.2VB -0.6x -0.6(X+2)

Thank You..!!